It defintely does. Already since Waddington, the very idea of the epigenetic landscape recognizes the importance of culture and nurture over the genes.
Currently there is no doubt about the interplay and influence of upbringing, among many other facttors, over the genetic code. Medically speaking, that is known as the allostatic overload.
Genetic memory is in genotype,it's transmitted and reproduced by inheritance.It's the only type of memory,which can't be influenced with upbringing."Brain aging is conclusively linked to genes; a crucial first step in finding biological mechanisms of normal aging"(Texas Biomedical Research Institute and Yale University)".White matter actively affects how the brain learns and functions (new information perception, the process of learning,thinking and memory)."Genetic material shared amongst biological relatives appears to predict the observed changes in brain function with age.By applying a sophisticated analysis the scientists demonstrated a heritable basis for neurocognitive deterioration with age that could be attributed to genetic factors."But a brain has plasticity; upbringing can change it more effective than aging.